A VARIATIONAL ITERATION METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS USING THE ABOODH TRANSFORM

In this article we propose a modification of the Variational Iteration Method (VIM) by adopting Aboudh transformation method. With this method a Lagrange multiplier is determined and the technique enhances the anatical solution of both the linear and nonlinear ordinary differential equations. Adoption of the Adomian series is also demonstrated. Some examples are given to illustrate the process.


INTRODUCTION
The variational iteration method (VIM) is an analytical technique that has been widely used in the last ten years [3,11,12], and the results in the approximate solutions have a high accuracy.Many researchers are making efforts for its developments [5,6,10].In this article, the transformation of Aboodh is adopted in some know analytical methods, the combination of Aboudh and the homotopy perturbation method is used to simplify the solution process and improve the accuracy of the solution.We give a new modified variational iteration method, based on the theory of variational iteration and Aboudh transform.In this work, we will use this method to solve some ordinary differential equations, including non-linear ones and differential equations with variable coefficients.The transformation of Aboodh and the non-linear term will be treated by the Adomian series.The results show that this new modification scheme.

THE ABOODH TRANSFORM
The Aboodh transform is a new integral transform similar to the Laplace transform and other integral transforms that are defined in the time domain 0 t ≥ , such as the Sumudu transform, the Natural transform and the Elzaki transform, respectively.An Aboodh transform is defined for functions of exponential order.We consider functions in the set F defined by : For a given function in the set F , M must be finite number and 1 k , 2 k , may be finite or infinite with variable v defined as . Then, the Aboodh transform denoted by the operator (:) A is defined by the integral equation: TABLE1: Aboodh transform of some functions.
Theorem1: Aboodh transform of some partial derivatives: The Aboodh transform is linear, i.e., if α and β are any constants, ( ) f t and ( ) g t are functions defined over the set F above, then Inspired by Wu's[3] method, if we take Aboodh transform on both sides of Eqs ( 4) and ( 5), the linear part with constant coefficients is then transferred into an algebraic one, so that we can identify the Lagrange multiplier in a more straightforward way.Now, we extend this idea to find the unknown Lagrange multiplier.
Appling the Aboodh's transform to eq (4) ,and using the results of eq (3), we have eqn (6): where W(v)=A(w(t)) , The iteration formula of eq(8) can be used to suggest the main iterative scheme involving the Lagrange multiplier as as restricted terms, and taking the classical variation operator on both sides of eq( 9) the iteration formula eq(9) can be explicitly given as where 0 w ( ) t is an initial approximation of Eq(4), and

Illustrative Examples
In this section, we apply the above defined method to solve both ordinary differential equations .

Linear and Nonlinear ODEs:
Exemple1: We consider the following simple linear first order differential equation In form general, we have [ ] Using the above procedure and employing (9) leads to the iterative formula and the iteration formula , involving the Lagrange multiplier where ( ) is an initial approximation of Eq (16).Let , where i B is the Adomian polynomials, then the Adomian series of the term 2 u reads .

. (18
with symbolic computation, we can derive the following approximate solutions ( ) ( ) ( ) ( ) M so the exact solution is given by

Variable Coefficient Ordinary Equation
Now, consider the following linear equation( 4) with variable coefficients and assume the linear term as where 1 2 , Q Q are also the linear operator, a is a constant and ( ) b t is a variable coefficient, then the new modified VIM is given as follows: Takinging the Aboodh transform on both sides of eq(4) leads to the iterative formula  as restricted terms, the Lagrange multiplier will be easily obtain, then repeat the above iteration process.
Example 3. Consider the following differential equations with a variable coefficient Applying the Aboodh transform on both sides of eq(21), we have ( ) A tw t is restricted variations, take the classical variation operator on both sides of eq (22) [ ] 1 ( ) ( 1) ( )....( 23) a Lagrange multiplier can be determined as  [ ] where 0 ( ) w t s an initial approximation of eq (21).Then, we have after a series of computation, we can derive the following approximate solutions

CONCLUSION
In this paper, exact solutions of nonlinear differential equations are obtained using Aboodh transformation and differential transformation methods.This method is more efficient and easy to manage such differential equations.The results reveal that this method is very efficient, simple and can be applied to other nonlinear problems.

.
Applying the procedure in sub section (2.2) we get the successive formula [ ] Consider the following first oeder nonlinear logistic differential equation:

A
− , the iteration formula eq (22) can be explicitly given as A VARIATIONAL ITERATION METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS [ ]